Well, using animal terms about adult people is usually insulting, so I'm going to say it's more like 'guzzle down' or 'lap up' (like a cat) and refers to someone drinking at the slightest provocation (not to mark a special occasion or relief after a long hard day).

I don't think it's right but does it mean to have a drink? An alcoholic one?

Yes, exactly, and in the sense of long binge when said: moczyć ryja instead of umoczyć ryja. Taken from a patriotic film about two dipsos and a history teacher. hahaha. When they start boozing, the teacher proves a bigger drinker than the two.

As for twisting sth`s head, it means to finish sth definitely - to break the neck of a problem, sort of. I was thinking about other verbs instead of twist but decided to keep it as that is the original version.